Problems of Control and Information Theory 8. (Budapest, 1979)

Problems of Control and Information Theory, Vol. 8(1), pp. 67 — 73 (1979)
NONPARAMETRIC SYSTEM IDENTIFICATION BY ORTHOGONAL SERIES
W. GREBLICKI (Wroclaw)
(Received February 20, 1977)
A nonparametric statistical method for memoryless system identification is presented. The method, based on the expansion in an infinite orthogonal series is asymptotically optimal. This means that a sequence of models converges to that one, which is optimal among all measurable models, i.e. converges to the regression of the output on the input. The square quality index converges to that of the optimal model.
1. Introduction
Almost all papers concerning statistical identification of memoryless systems assume that a model is described by finite number of parameters, see Aizerman [
1], Bubnicki [2], Saridis [6], Tsypkin [8]. A special attention is given to the stochastic approximation, see [3], [6], [8]. The nonparametric approach is presented by Braverman [2], and the problem is treated by the method of potential functions. Another nonparametric method for identifi­cation by orthogonal series is given in this lecture and its basic asymptotic properties are shown.
2. Statement of the problem
The input of a memoryless system is a Ldimentional vector x which belongs to a set % d Jtfc, the output is a scalar у £ ^ = át1. Let у be a ff-finite measure on the space át,f, and let % be /«-measurable. Both the input and the output are random. We denote the input-output pair of random variables by (X, Y) and assume that EY2 < oo. Let f(x, y) be its probability density function f(yjx) is the conditional output density and f(x) is the input density. Obviously, the conditional output density describes the system’s properties.
Any /«-measurable function Ф(х) mapping % into D will be called a model. It is known that the quality index
6*
ЩФ) = JJ (y - 0(x))2f(x> y)dxdy
(1)